Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

The important point of the proof either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.

EDIT: Brian Conrad points out below that while this is a fairly complete sketch in the smooth case, it requires more work in the singular case.

You might also want to read Carlos Simpson's paper "Moduli of representations of the fundamental group of a smooth projective variety", parts I and II. He explains in great detail how to make the set of objects of each of these categories into the points of an algebraic variety, and why these algebraic varieties are analytically but not algebraically isomorphic. He doesn't use the category theoretic perspective much, as I recall, but he is invaluable for understanding how to work with concrete moduli spaces.

More explicitly, given a local system $\mathcal{V}$ you take the vector bundle to be $\mathcal{E}=\mathcal{V} \otimes_{\mathbb{C}} \mathcal{O}_X$, where $\mathcal{O}_X$ is the structure sheaf, and use $d: \mathcal{O}_X \to \Omega_X$ to define the flat connection on $\mathcal{E}$. Conversely, given $D:\mathcal{E} \to \mathcal{E} \otimes\Omega_X$ you take $\mathcal{V}$ to be $\text{ker}(D)$.

And if the analytic space is connected, one can add one more equivalence: once we choose a point $x$ on the space (choosing a "fiber functor"), those two categories are equivalent to complex representations of $\pi_1(X,x)$ (the "Tannaka dual").